Properties

Label 201.4.j.a.137.2
Level $201$
Weight $4$
Character 201.137
Analytic conductor $11.859$
Analytic rank $0$
Dimension $20$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(5,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label 137.2
Root \(0.0475819 + 0.998867i\) of defining polynomial
Character \(\chi\) \(=\) 201.137
Dual form 201.4.j.a.179.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.14326 + 0.739490i) q^{3} +(-6.73003 - 4.32513i) q^{4} +(-4.34287 + 14.7905i) q^{7} +(25.9063 + 7.60678i) q^{9} +O(q^{10})\) \(q+(5.14326 + 0.739490i) q^{3} +(-6.73003 - 4.32513i) q^{4} +(-4.34287 + 14.7905i) q^{7} +(25.9063 + 7.60678i) q^{9} +(-31.4159 - 27.2220i) q^{12} +(60.1194 + 52.0937i) q^{13} +(26.5866 + 58.2164i) q^{16} +(140.401 - 41.2254i) q^{19} +(-33.2739 + 72.8597i) q^{21} +(81.8576 - 94.4687i) q^{25} +(127.618 + 58.2811i) q^{27} +(93.1983 - 80.7568i) q^{28} +(-162.796 + 141.063i) q^{31} +(-141.450 - 163.242i) q^{36} -257.832 q^{37} +(270.687 + 312.389i) q^{39} +(135.200 + 210.376i) q^{43} +(93.6912 + 319.083i) q^{48} +(88.6527 + 56.9736i) q^{49} +(-179.293 - 610.616i) q^{52} +(752.605 - 108.208i) q^{57} +(-42.7885 - 19.5409i) q^{61} +(-225.016 + 350.131i) q^{63} +(72.8652 - 506.789i) q^{64} +(-40.7384 + 546.903i) q^{67} +(-63.9862 + 140.110i) q^{73} +(490.874 - 425.345i) q^{75} +(-1123.21 - 329.803i) q^{76} +(-1044.95 - 905.451i) q^{79} +(613.274 + 394.127i) q^{81} +(539.062 - 346.434i) q^{84} +(-1031.58 + 662.957i) q^{91} +(-941.617 + 605.141i) q^{93} -1121.24i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{22}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(3\) 5.14326 + 0.739490i 0.989821 + 0.142315i
\(4\) −6.73003 4.32513i −0.841254 0.540641i
\(5\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(6\) 0 0
\(7\) −4.34287 + 14.7905i −0.234493 + 0.798610i 0.755211 + 0.655482i \(0.227534\pi\)
−0.989704 + 0.143128i \(0.954284\pi\)
\(8\) 0 0
\(9\) 25.9063 + 7.60678i 0.959493 + 0.281733i
\(10\) 0 0
\(11\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(12\) −31.4159 27.2220i −0.755750 0.654861i
\(13\) 60.1194 + 52.0937i 1.28262 + 1.11140i 0.987780 + 0.155855i \(0.0498133\pi\)
0.294844 + 0.955545i \(0.404732\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 26.5866 + 58.2164i 0.415415 + 0.909632i
\(17\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(18\) 0 0
\(19\) 140.401 41.2254i 1.69527 0.497777i 0.715622 0.698487i \(-0.246143\pi\)
0.979651 + 0.200710i \(0.0643250\pi\)
\(20\) 0 0
\(21\) −33.2739 + 72.8597i −0.345760 + 0.757109i
\(22\) 0 0
\(23\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(24\) 0 0
\(25\) 81.8576 94.4687i 0.654861 0.755750i
\(26\) 0 0
\(27\) 127.618 + 58.2811i 0.909632 + 0.415415i
\(28\) 93.1983 80.7568i 0.629029 0.545057i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −162.796 + 141.063i −0.943194 + 0.817282i −0.983315 0.181913i \(-0.941771\pi\)
0.0401208 + 0.999195i \(0.487226\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −141.450 163.242i −0.654861 0.755750i
\(37\) −257.832 −1.14561 −0.572803 0.819693i \(-0.694144\pi\)
−0.572803 + 0.819693i \(0.694144\pi\)
\(38\) 0 0
\(39\) 270.687 + 312.389i 1.11140 + 1.28262i
\(40\) 0 0
\(41\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(42\) 0 0
\(43\) 135.200 + 210.376i 0.479485 + 0.746093i 0.993761 0.111532i \(-0.0355757\pi\)
−0.514276 + 0.857625i \(0.671939\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(48\) 93.6912 + 319.083i 0.281733 + 0.959493i
\(49\) 88.6527 + 56.9736i 0.258463 + 0.166104i
\(50\) 0 0
\(51\) 0 0
\(52\) −179.293 610.616i −0.478144 1.62841i
\(53\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 752.605 108.208i 1.74886 0.251448i
\(58\) 0 0
\(59\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(60\) 0 0
\(61\) −42.7885 19.5409i −0.0898116 0.0410156i 0.370003 0.929031i \(-0.379357\pi\)
−0.459814 + 0.888015i \(0.652084\pi\)
\(62\) 0 0
\(63\) −225.016 + 350.131i −0.449989 + 0.700196i
\(64\) 72.8652 506.789i 0.142315 0.989821i
\(65\) 0 0
\(66\) 0 0
\(67\) −40.7384 + 546.903i −0.0742833 + 0.997237i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(72\) 0 0
\(73\) −63.9862 + 140.110i −0.102589 + 0.224639i −0.953966 0.299916i \(-0.903041\pi\)
0.851376 + 0.524556i \(0.175769\pi\)
\(74\) 0 0
\(75\) 490.874 425.345i 0.755750 0.654861i
\(76\) −1123.21 329.803i −1.69527 0.497777i
\(77\) 0 0
\(78\) 0 0
\(79\) −1044.95 905.451i −1.48817 1.28951i −0.858692 0.512491i \(-0.828723\pi\)
−0.629480 0.777017i \(-0.716732\pi\)
\(80\) 0 0
\(81\) 613.274 + 394.127i 0.841254 + 0.540641i
\(82\) 0 0
\(83\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(84\) 539.062 346.434i 0.700196 0.449989i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(90\) 0 0
\(91\) −1031.58 + 662.957i −1.18834 + 0.763701i
\(92\) 0 0
\(93\) −941.617 + 605.141i −1.04990 + 0.674733i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1121.24i 1.17365i −0.809712 0.586827i \(-0.800377\pi\)
0.809712 0.586827i \(-0.199623\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −959.493 + 281.733i −0.959493 + 0.281733i
\(101\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(102\) 0 0
\(103\) −1333.38 1538.81i −1.27556 1.47207i −0.809271 0.587436i \(-0.800137\pi\)
−0.466285 0.884634i \(-0.654408\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(108\) −606.798 944.197i −0.540641 0.841254i
\(109\) −1129.25 978.504i −0.992320 0.859851i −0.00218975 0.999998i \(-0.500697\pi\)
−0.990131 + 0.140147i \(0.955242\pi\)
\(110\) 0 0
\(111\) −1326.10 190.664i −1.13394 0.163037i
\(112\) −976.510 + 140.401i −0.823853 + 0.118452i
\(113\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1161.21 + 1806.87i 0.917551 + 1.42774i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 871.620 1005.90i 0.654861 0.755750i
\(122\) 0 0
\(123\) 0 0
\(124\) 1705.74 245.248i 1.23532 0.177612i
\(125\) 0 0
\(126\) 0 0
\(127\) 1818.52 + 533.965i 1.27061 + 0.373084i 0.846433 0.532496i \(-0.178746\pi\)
0.424176 + 0.905580i \(0.360564\pi\)
\(128\) 0 0
\(129\) 539.800 + 1182.00i 0.368424 + 0.806736i
\(130\) 0 0
\(131\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(132\) 0 0
\(133\) 2255.63i 1.47059i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(138\) 0 0
\(139\) 2581.70 1179.02i 1.57537 0.719449i 0.579918 0.814675i \(-0.303085\pi\)
0.995456 + 0.0952257i \(0.0303573\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 245.920 + 1710.41i 0.142315 + 0.989821i
\(145\) 0 0
\(146\) 0 0
\(147\) 413.833 + 358.588i 0.232193 + 0.201196i
\(148\) 1735.22 + 1115.16i 0.963745 + 0.619361i
\(149\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(150\) 0 0
\(151\) 3115.91 2002.48i 1.67927 1.07920i 0.806106 0.591771i \(-0.201571\pi\)
0.873163 0.487429i \(-0.162065\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −470.607 3273.15i −0.241530 1.67988i
\(157\) −558.219 + 3882.50i −0.283762 + 1.97361i −0.0646697 + 0.997907i \(0.520599\pi\)
−0.219093 + 0.975704i \(0.570310\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4158.14 −1.99810 −0.999050 0.0435727i \(-0.986126\pi\)
−0.999050 + 0.0435727i \(0.986126\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(168\) 0 0
\(169\) 587.916 + 4089.05i 0.267600 + 1.86120i
\(170\) 0 0
\(171\) 3950.86 1.76684
\(172\) 2000.59i 0.886882i
\(173\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(174\) 0 0
\(175\) 1041.74 + 1620.98i 0.449989 + 0.700196i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(180\) 0 0
\(181\) 553.132 3847.12i 0.227149 1.57986i −0.482882 0.875686i \(-0.660410\pi\)
0.710031 0.704171i \(-0.248681\pi\)
\(182\) 0 0
\(183\) −205.622 132.145i −0.0830603 0.0533796i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1416.23 + 1634.42i −0.545057 + 0.629029i
\(190\) 0 0
\(191\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(192\) 749.530 2552.66i 0.281733 0.959493i
\(193\) 243.752 + 281.305i 0.0909102 + 0.104916i 0.799382 0.600823i \(-0.205160\pi\)
−0.708472 + 0.705739i \(0.750615\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −350.217 766.868i −0.127630 0.279471i
\(197\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(198\) 0 0
\(199\) −3175.15 932.308i −1.13106 0.332108i −0.337934 0.941170i \(-0.609728\pi\)
−0.793123 + 0.609061i \(0.791546\pi\)
\(200\) 0 0
\(201\) −613.957 + 2782.74i −0.215449 + 0.976515i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1434.34 + 4884.93i −0.478144 + 1.62841i
\(209\) 0 0
\(210\) 0 0
\(211\) 497.869 + 3462.76i 0.162440 + 1.12979i 0.894017 + 0.448033i \(0.147875\pi\)
−0.731577 + 0.681759i \(0.761215\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1379.39 3020.45i −0.431517 0.944891i
\(218\) 0 0
\(219\) −432.708 + 673.307i −0.133515 + 0.207753i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 756.046 + 5258.41i 0.227034 + 1.57906i 0.710503 + 0.703694i \(0.248468\pi\)
−0.483469 + 0.875362i \(0.660623\pi\)
\(224\) 0 0
\(225\) 2839.23 1824.66i 0.841254 0.540641i
\(226\) 0 0
\(227\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(228\) −5533.07 2526.87i −1.60718 0.733973i
\(229\) 5173.85 4483.17i 1.49300 1.29370i 0.644370 0.764714i \(-0.277120\pi\)
0.848633 0.528981i \(-0.177426\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4704.86 5429.70i −1.28951 1.48817i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1615.58 3537.64i 0.431821 0.945557i −0.561206 0.827676i \(-0.689663\pi\)
0.993028 0.117881i \(-0.0376101\pi\)
\(242\) 0 0
\(243\) 2862.78 + 2480.61i 0.755750 + 0.654861i
\(244\) 203.451 + 316.576i 0.0533796 + 0.0830603i
\(245\) 0 0
\(246\) 0 0
\(247\) 10588.4 + 4835.56i 2.72763 + 1.24567i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(252\) 3028.72 1383.17i 0.757109 0.345760i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −2682.31 + 3095.55i −0.654861 + 0.755750i
\(257\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(258\) 0 0
\(259\) 1119.73 3813.46i 0.268636 0.914892i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2639.60 3504.48i 0.601638 0.798769i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −8212.28 1180.75i −1.84081 0.264669i −0.868052 0.496474i \(-0.834628\pi\)
−0.972762 + 0.231804i \(0.925537\pi\)
\(272\) 0 0
\(273\) −5795.94 + 2646.92i −1.28493 + 0.586809i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1049.03 + 308.022i 0.227545 + 0.0668132i 0.393517 0.919317i \(-0.371258\pi\)
−0.165972 + 0.986130i \(0.553076\pi\)
\(278\) 0 0
\(279\) −5290.48 + 2416.08i −1.13524 + 0.518448i
\(280\) 0 0
\(281\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(282\) 0 0
\(283\) −8953.15 + 2628.88i −1.88060 + 0.552194i −0.884230 + 0.467052i \(0.845316\pi\)
−0.996369 + 0.0851419i \(0.972866\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2040.93 4469.02i 0.415415 0.909632i
\(290\) 0 0
\(291\) 829.143 5766.82i 0.167028 1.16171i
\(292\) 1036.62 666.198i 0.207753 0.133515i
\(293\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −5143.26 + 739.490i −0.989821 + 0.142315i
\(301\) −3698.71 + 1086.04i −0.708273 + 0.207968i
\(302\) 0 0
\(303\) 0 0
\(304\) 6132.78 + 7077.60i 1.15704 + 1.33529i
\(305\) 0 0
\(306\) 0 0
\(307\) −6424.56 7414.33i −1.19436 1.37837i −0.907316 0.420449i \(-0.861873\pi\)
−0.287045 0.957917i \(-0.592673\pi\)
\(308\) 0 0
\(309\) −5720.01 8900.52i −1.05308 1.63862i
\(310\) 0 0
\(311\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(312\) 0 0
\(313\) −7292.09 + 1048.44i −1.31685 + 0.189334i −0.764686 0.644403i \(-0.777106\pi\)
−0.552161 + 0.833737i \(0.686197\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3116.33 + 10613.2i 0.554769 + 1.88937i
\(317\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −2422.70 5304.97i −0.415415 0.909632i
\(325\) 9842.46 1415.13i 1.67988 0.241530i
\(326\) 0 0
\(327\) −5084.46 5867.78i −0.859851 0.992320i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1454.39 + 2263.08i −0.241512 + 0.375800i −0.940757 0.339082i \(-0.889884\pi\)
0.699245 + 0.714883i \(0.253520\pi\)
\(332\) 0 0
\(333\) −6679.49 1961.27i −1.09920 0.322754i
\(334\) 0 0
\(335\) 0 0
\(336\) −5126.27 −0.832325
\(337\) 1044.04 3555.66i 0.168761 0.574746i −0.831067 0.556173i \(-0.812269\pi\)
0.999827 0.0185734i \(-0.00591244\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −5223.55 + 4526.23i −0.822289 + 0.712518i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(348\) 0 0
\(349\) −8360.54 5372.99i −1.28232 0.824097i −0.291148 0.956678i \(-0.594037\pi\)
−0.991172 + 0.132581i \(0.957673\pi\)
\(350\) 0 0
\(351\) 4636.22 + 10151.9i 0.705024 + 1.54379i
\(352\) 0 0
\(353\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(360\) 0 0
\(361\) 12242.7 7867.92i 1.78491 1.14709i
\(362\) 0 0
\(363\) 5226.82 4529.07i 0.755750 0.654861i
\(364\) 9809.94 1.41258
\(365\) 0 0
\(366\) 0 0
\(367\) 9407.20 1352.55i 1.33802 0.192378i 0.564121 0.825692i \(-0.309215\pi\)
0.773895 + 0.633314i \(0.218306\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 8954.42 1.24802
\(373\) 10598.9i 1.47129i 0.677366 + 0.735646i \(0.263121\pi\)
−0.677366 + 0.735646i \(0.736879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 13793.5 + 1983.20i 1.86945 + 0.268787i 0.981544 0.191236i \(-0.0612496\pi\)
0.887907 + 0.460022i \(0.152159\pi\)
\(380\) 0 0
\(381\) 8958.25 + 4091.10i 1.20458 + 0.550113i
\(382\) 0 0
\(383\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1902.26 + 6478.50i 0.249864 + 0.850957i
\(388\) −4849.49 + 7545.96i −0.634525 + 0.987340i
\(389\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3958.90 8668.78i −0.500482 1.09590i −0.976312 0.216366i \(-0.930580\pi\)
0.475830 0.879537i \(-0.342148\pi\)
\(398\) 0 0
\(399\) −1668.02 + 11601.3i −0.209286 + 1.45562i
\(400\) 7675.94 + 2253.86i 0.959493 + 0.281733i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −17135.7 −2.11809
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3791.68 12913.3i 0.458403 1.56118i −0.328749 0.944417i \(-0.606627\pi\)
0.787152 0.616759i \(-0.211555\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2318.18 + 16123.3i 0.277205 + 1.92800i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14150.2 4154.88i 1.66173 0.487927i
\(418\) 0 0
\(419\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(420\) 0 0
\(421\) −12350.9 + 3626.56i −1.42980 + 0.419828i −0.902809 0.430041i \(-0.858499\pi\)
−0.526994 + 0.849869i \(0.676681\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 474.843 547.999i 0.0538157 0.0621066i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 8978.95i 1.00000i
\(433\) 10305.1 8929.45i 1.14373 0.991044i 0.143727 0.989617i \(-0.454091\pi\)
0.999999 0.00142668i \(-0.000454128\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3367.76 + 11469.5i 0.369923 + 1.25984i
\(437\) 0 0
\(438\) 0 0
\(439\) −18370.0 −1.99716 −0.998580 0.0532736i \(-0.983034\pi\)
−0.998580 + 0.0532736i \(0.983034\pi\)
\(440\) 0 0
\(441\) 1863.28 + 2150.34i 0.201196 + 0.232193i
\(442\) 0 0
\(443\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(444\) 8100.04 + 7018.73i 0.865791 + 0.750212i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 7179.19 + 3278.63i 0.757109 + 0.345760i
\(449\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 17506.8 7995.08i 1.81576 0.829231i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10724.1 12376.3i 1.09771 1.26682i 0.136604 0.990626i \(-0.456381\pi\)
0.961102 0.276194i \(-0.0890732\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(462\) 0 0
\(463\) 7164.31 + 3271.83i 0.719122 + 0.328412i 0.741145 0.671345i \(-0.234283\pi\)
−0.0220224 + 0.999757i \(0.507011\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(468\) 17182.7i 1.69716i
\(469\) −7912.03 2977.67i −0.778985 0.293169i
\(470\) 0 0
\(471\) −5742.13 + 19555.9i −0.561748 + 1.91314i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7598.37 16638.1i 0.733973 1.60718i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(480\) 0 0
\(481\) −15500.7 13431.5i −1.46938 1.27323i
\(482\) 0 0
\(483\) 0 0
\(484\) −10216.7 + 2999.89i −0.959493 + 0.281733i
\(485\) 0 0
\(486\) 0 0
\(487\) −8391.85 + 13058.0i −0.780844 + 1.21502i 0.191506 + 0.981491i \(0.438663\pi\)
−0.972351 + 0.233526i \(0.924974\pi\)
\(488\) 0 0
\(489\) −21386.4 3074.90i −1.97776 0.284359i
\(490\) 0 0
\(491\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −12540.4 5727.01i −1.13524 0.518448i
\(497\) 0 0
\(498\) 0 0
\(499\) 22157.7i 1.98781i 0.110258 + 0.993903i \(0.464832\pi\)
−0.110258 + 0.993903i \(0.535168\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 21465.8i 1.88034i
\(508\) −9929.21 11458.9i −0.867199 1.00080i
\(509\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(510\) 0 0
\(511\) −1794.41 1554.87i −0.155343 0.134605i
\(512\) 0 0
\(513\) 20320.3 + 2921.62i 1.74886 + 0.251448i
\(514\) 0 0
\(515\) 0 0
\(516\) 1479.42 10289.6i 0.126216 0.877855i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(522\) 0 0
\(523\) 3751.57 4329.54i 0.313661 0.361984i −0.576927 0.816796i \(-0.695748\pi\)
0.890587 + 0.454812i \(0.150294\pi\)
\(524\) 0 0
\(525\) 4159.24 + 9107.47i 0.345760 + 0.757109i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11674.2 3427.84i −0.959493 0.281733i
\(530\) 0 0
\(531\) 0 0
\(532\) 9755.89 15180.5i 0.795059 1.23714i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2802.54 + 1279.88i −0.222718 + 0.101712i −0.523649 0.851934i \(-0.675430\pi\)
0.300931 + 0.953646i \(0.402703\pi\)
\(542\) 0 0
\(543\) 5689.81 19377.7i 0.449674 1.53145i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18732.8 8554.98i 1.46427 0.668711i 0.485608 0.874177i \(-0.338598\pi\)
0.978664 + 0.205466i \(0.0658710\pi\)
\(548\) 0 0
\(549\) −959.850 831.714i −0.0746182 0.0646570i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 17930.1 11523.0i 1.37878 0.886088i
\(554\) 0 0
\(555\) 0 0
\(556\) −22474.3 3231.32i −1.71425 0.246472i
\(557\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(558\) 0 0
\(559\) −2831.10 + 19690.7i −0.214209 + 1.48986i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −8492.69 + 7358.96i −0.629029 + 0.545057i
\(568\) 0 0
\(569\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(570\) 0 0
\(571\) 2470.41 + 17182.1i 0.181057 + 1.25928i 0.854270 + 0.519829i \(0.174004\pi\)
−0.673214 + 0.739448i \(0.735087\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 5742.70 12574.8i 0.415415 0.909632i
\(577\) −13498.9 21004.7i −0.973945 1.51549i −0.852393 0.522902i \(-0.824849\pi\)
−0.121552 0.992585i \(-0.538787\pi\)
\(578\) 0 0
\(579\) 1045.66 + 1627.08i 0.0750537 + 0.116786i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(588\) −1234.17 4203.19i −0.0865582 0.294790i
\(589\) −17041.3 + 26516.8i −1.19215 + 1.85502i
\(590\) 0 0
\(591\) 0 0
\(592\) −6854.88 15010.1i −0.475902 1.04208i
\(593\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −15641.2 7143.10i −1.07228 0.489694i
\(598\) 0 0
\(599\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(600\) 0 0
\(601\) −8958.34 2630.41i −0.608017 0.178530i −0.0367916 0.999323i \(-0.511714\pi\)
−0.571226 + 0.820793i \(0.693532\pi\)
\(602\) 0 0
\(603\) −5215.55 + 13858.4i −0.352228 + 0.935914i
\(604\) −29631.2 −1.99615
\(605\) 0 0
\(606\) 0 0
\(607\) 20730.6 + 13322.7i 1.38621 + 0.890863i 0.999509 0.0313428i \(-0.00997836\pi\)
0.386701 + 0.922205i \(0.373615\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3996.21 27794.3i −0.263304 1.83132i −0.507566 0.861613i \(-0.669455\pi\)
0.244262 0.969709i \(-0.421454\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(618\) 0 0
\(619\) −10431.6 22842.0i −0.677353 1.48320i −0.865424 0.501040i \(-0.832951\pi\)
0.188072 0.982155i \(-0.439776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −10989.6 + 24063.8i −0.705024 + 1.54379i
\(625\) −2223.67 15466.0i −0.142315 0.989821i
\(626\) 0 0
\(627\) 0 0
\(628\) 20549.1 23714.9i 1.30573 1.50689i
\(629\) 0 0
\(630\) 0 0
\(631\) −1988.19 + 1722.77i −0.125433 + 0.108689i −0.715319 0.698798i \(-0.753718\pi\)
0.589885 + 0.807487i \(0.299173\pi\)
\(632\) 0 0
\(633\) 18178.0i 1.14141i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2361.78 + 8043.47i 0.146903 + 0.500304i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −11300.0 + 24743.5i −0.693044 + 1.51755i 0.155163 + 0.987889i \(0.450410\pi\)
−0.848207 + 0.529665i \(0.822318\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4860.99 16555.0i −0.292653 0.996685i
\(652\) 27984.4 + 17984.5i 1.68091 + 1.08025i
\(653\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2723.43 + 3143.01i −0.161722 + 0.186637i
\(658\) 0 0
\(659\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(660\) 0 0
\(661\) 3308.86 11268.9i 0.194705 0.663103i −0.803037 0.595930i \(-0.796784\pi\)
0.997741 0.0671734i \(-0.0213981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 27604.5i 1.59529i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 11244.5 + 1616.72i 0.644048 + 0.0926001i 0.456597 0.889674i \(-0.349068\pi\)
0.187451 + 0.982274i \(0.439977\pi\)
\(674\) 0 0
\(675\) 15952.2 7285.14i 0.909632 0.415415i
\(676\) 13729.0 30062.2i 0.781120 1.71041i
\(677\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(678\) 0 0
\(679\) 16583.6 + 4869.39i 0.937291 + 0.275214i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(684\) −26589.4 17088.0i −1.48636 0.955227i
\(685\) 0 0
\(686\) 0 0
\(687\) 29925.7 19232.1i 1.66192 1.06805i
\(688\) −8652.81 + 13464.0i −0.479485 + 0.746093i
\(689\) 0 0
\(690\) 0 0
\(691\) −14600.7 + 31971.0i −0.803813 + 1.76011i −0.171826 + 0.985127i \(0.554967\pi\)
−0.631988 + 0.774978i \(0.717761\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 15414.9i 0.832325i
\(701\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(702\) 0 0
\(703\) −36199.9 + 10629.3i −1.94211 + 0.570256i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10441.7 12050.4i −0.553099 0.638311i 0.408503 0.912757i \(-0.366051\pi\)
−0.961602 + 0.274446i \(0.911505\pi\)
\(710\) 0 0
\(711\) −20183.1 31405.6i −1.06459 1.65654i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(720\) 0 0
\(721\) 28550.4 13038.5i 1.47472 0.673482i
\(722\) 0 0
\(723\) 10925.4 17000.3i 0.561993 0.874478i
\(724\) −20361.9 + 23498.8i −1.04522 + 1.20625i
\(725\) 0 0
\(726\) 0 0
\(727\) 25617.7 3683.27i 1.30689 0.187902i 0.546549 0.837427i \(-0.315941\pi\)
0.760341 + 0.649524i \(0.225032\pi\)
\(728\) 0 0
\(729\) 12889.6 + 14875.4i 0.654861 + 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) 812.298 + 1778.69i 0.0410156 + 0.0898116i
\(733\) −16759.1 + 26077.7i −0.844492 + 1.31406i 0.103132 + 0.994668i \(0.467114\pi\)
−0.947624 + 0.319388i \(0.896523\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1169.23 + 3982.03i −0.0582014 + 0.198216i −0.983461 0.181118i \(-0.942029\pi\)
0.925260 + 0.379333i \(0.123847\pi\)
\(740\) 0 0
\(741\) 50883.1 + 32700.6i 2.52259 + 1.62117i
\(742\) 0 0
\(743\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1212.12 778.981i −0.0588959 0.0378501i 0.510862 0.859663i \(-0.329326\pi\)
−0.569757 + 0.821813i \(0.692963\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 16600.4 4874.30i 0.798610 0.234493i
\(757\) 38953.7 + 5600.70i 1.87027 + 0.268905i 0.981785 0.189993i \(-0.0608465\pi\)
0.888489 + 0.458898i \(0.151756\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(762\) 0 0
\(763\) 19376.7 12452.7i 0.919377 0.590848i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −16085.0 + 13937.7i −0.755750 + 0.654861i
\(769\) −24038.3 + 3456.19i −1.12724 + 0.162072i −0.680608 0.732648i \(-0.738284\pi\)
−0.446627 + 0.894720i \(0.647375\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −423.779 2947.45i −0.0197567 0.137411i
\(773\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(774\) 0 0
\(775\) 26926.2i 1.24802i
\(776\) 0 0
\(777\) 8579.10 18785.6i 0.396105 0.867349i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −959.832 + 6675.78i −0.0437241 + 0.304108i
\(785\) 0 0
\(786\) 0 0
\(787\) 20613.7 + 32075.5i 0.933670 + 1.45282i 0.891142 + 0.453724i \(0.149905\pi\)
0.0425282 + 0.999095i \(0.486459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1554.46 3403.80i −0.0696098 0.152424i
\(794\) 0 0
\(795\) 0 0
\(796\) 17336.5 + 20007.4i 0.771955 + 0.890883i
\(797\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 16167.7 16072.5i 0.709191 0.705016i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(810\) 0 0
\(811\) −12840.0 + 43729.1i −0.555948 + 1.89338i −0.121767 + 0.992559i \(0.538856\pi\)
−0.434181 + 0.900826i \(0.642962\pi\)
\(812\) 0 0
\(813\) −41364.8 12145.8i −1.78441 0.523950i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 27655.1 + 23963.3i 1.18425 + 1.02615i
\(818\) 0 0
\(819\) −31767.4 + 9327.76i −1.35536 + 0.397971i
\(820\) 0 0
\(821\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(822\) 0 0
\(823\) −1079.75 + 317.043i −0.0457323 + 0.0134282i −0.304519 0.952506i \(-0.598496\pi\)
0.258786 + 0.965935i \(0.416677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(828\) 0 0
\(829\) 25186.1 29066.3i 1.05519 1.21775i 0.0798996 0.996803i \(-0.474540\pi\)
0.975286 0.220946i \(-0.0709145\pi\)
\(830\) 0 0
\(831\) 5167.65 + 2359.98i 0.215720 + 0.0985162i
\(832\) 30781.1 26672.0i 1.28262 1.11140i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −28997.0 + 8514.29i −1.19747 + 0.351609i
\(838\) 0 0
\(839\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(840\) 0 0
\(841\) 24389.0 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 11626.2 25457.8i 0.474159 1.03826i
\(845\) 0 0
\(846\) 0 0
\(847\) 11092.4 + 17260.2i 0.449989 + 0.700196i
\(848\) 0 0
\(849\) −47992.4 + 6900.27i −1.94004 + 0.278936i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 14661.3 + 9422.23i 0.588502 + 0.378207i 0.800741 0.599010i \(-0.204439\pi\)
−0.212239 + 0.977218i \(0.568076\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(858\) 0 0
\(859\) 32943.2 38018.4i 1.30851 1.51010i 0.621699 0.783256i \(-0.286443\pi\)
0.686807 0.726840i \(-0.259012\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13801.9 21476.1i 0.540641 0.841254i
\(868\) −3780.47 + 26293.7i −0.147831 + 1.02819i
\(869\) 0 0
\(870\) 0 0
\(871\) −30939.4 + 30757.3i −1.20361 + 1.19652i
\(872\) 0 0
\(873\) 8529.00 29047.1i 0.330656 1.12611i
\(874\) 0 0
\(875\) 0 0
\(876\) 5824.28 2659.86i 0.224639 0.102589i
\(877\) 18405.6 40302.6i 0.708680 1.55179i −0.120438 0.992721i \(-0.538430\pi\)
0.829119 0.559073i \(-0.188843\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(882\) 0 0
\(883\) 20657.2 + 17899.6i 0.787284 + 0.682185i 0.952657 0.304046i \(-0.0983376\pi\)
−0.165374 + 0.986231i \(0.552883\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(888\) 0 0
\(889\) −15795.2 + 24577.8i −0.595898 + 0.927235i
\(890\) 0 0
\(891\) 0 0
\(892\) 17655.1 38659.3i 0.662709 1.45113i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −27000.0 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −19826.6 + 2850.63i −0.730661 + 0.105053i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −30788.3 35531.6i −1.12713 1.30078i −0.948469 0.316871i \(-0.897368\pi\)
−0.178665 0.983910i \(-0.557178\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(912\) 26308.7 + 40937.1i 0.955227 + 1.48636i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −54210.5 + 7794.29i −1.95542 + 0.281147i
\(917\) 0 0
\(918\) 0 0
\(919\) −5806.85 19776.3i −0.208434 0.709860i −0.995649 0.0931816i \(-0.970296\pi\)
0.787216 0.616678i \(-0.211522\pi\)
\(920\) 0 0
\(921\) −27560.4 42884.8i −0.986042 1.53431i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −21105.5 + 24357.1i −0.750212 + 0.865791i
\(926\) 0 0
\(927\) −22837.7 50007.6i −0.809157 1.77181i
\(928\) 0 0
\(929\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(930\) 0 0
\(931\) 14795.7 + 4344.41i 0.520848 + 0.152935i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17841.3i 0.622039i 0.950404 + 0.311020i \(0.100670\pi\)
−0.950404 + 0.311020i \(0.899330\pi\)
\(938\) 0 0
\(939\) −38280.4 −1.33039
\(940\) 0 0
\(941\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(948\) 8179.71 + 56891.1i 0.280237 + 1.94909i
\(949\) −11145.7 + 5090.06i −0.381248 + 0.174110i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2363.91 16441.3i 0.0793497 0.551889i
\(962\) 0 0
\(963\) 0 0
\(964\) −26173.7 + 16820.8i −0.874478 + 0.561993i
\(965\) 0 0
\(966\) 0 0
\(967\) −40169.2 −1.33584 −0.667918 0.744234i \(-0.732815\pi\)
−0.667918 + 0.744234i \(0.732815\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(972\) −8537.61 29076.4i −0.281733 0.959493i
\(973\) 6226.30 + 43304.9i 0.205145 + 1.42681i
\(974\) 0 0
\(975\) 51668.8 1.69716
\(976\) 3010.52i 0.0987340i
\(977\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −21811.5 33939.4i −0.709877 1.10459i
\(982\) 0 0
\(983\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −50345.8 78339.7i −1.62117 2.52259i
\(989\) 0 0
\(990\) 0 0
\(991\) 31996.9 49788.2i 1.02565 1.59594i 0.246450 0.969155i \(-0.420736\pi\)
0.779196 0.626781i \(-0.215628\pi\)
\(992\) 0 0
\(993\) −9153.84 + 10564.1i −0.292536 + 0.337604i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33560.1 + 38730.5i 1.06606 + 1.23030i 0.972062 + 0.234725i \(0.0754189\pi\)
0.0939968 + 0.995572i \(0.470036\pi\)
\(998\) 0 0
\(999\) −32904.0 15026.8i −1.04208 0.475902i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 201.4.j.a.137.2 20
3.2 odd 2 CM 201.4.j.a.137.2 20
67.45 odd 22 inner 201.4.j.a.179.2 yes 20
201.179 even 22 inner 201.4.j.a.179.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.4.j.a.137.2 20 1.1 even 1 trivial
201.4.j.a.137.2 20 3.2 odd 2 CM
201.4.j.a.179.2 yes 20 67.45 odd 22 inner
201.4.j.a.179.2 yes 20 201.179 even 22 inner